2003 A simple model for spiking neurons

Eugene M. Izhikevich presented a model that reproduces spiking and bursting behavior of known types of cortical neurons [1]. The model combines the biologically plausibility of the dynamics underlying the Hodgkin–Huxley model [2] and the computational efficiency of integrate-and-fire neurons. As initiated by Bard Ermentrout and Nancy Kopell [3], this model is made of an oscillator producing slow oscillations combined with a switching mechanism for reproducing the bursting phenomenon [4]. The model equations - as proposed in Ref. [1] - are

where the switching mechanism is introduced as follows

when y>30.

Variable x represents the membrane recovery which accounts for the activation of K+ ionic currents and inactivation of Na+ ionic currents, and it provides negative feedback to the membrane potential of the neuron y. Synaptic currents or injected dc-currents are delivered via the variable Isyn. The
part 0.04y2+5y+140 was obtained by fitting the spike initiation dynamics of a cortical neuron so that the membrane potential is expressed in mV and the time in ms.

Using parameter values a=0.2, b=2, c=-56, d=-16 and Isyn=-99, and initial conditions as

a chaotic attractor can be obtained (Fig. 1). It is characterized by a first-return map to a Poincaré section made of four branches, more or less as can be found in the Rössler system [5].